Aerodynamic potential-flow code

Early panel codes were developed in the late 1960s to early 1970s. Advanced panel codes, such as Panair (developed by Boeing), were first introduced in the late 1970s, and gained popularity as computing speed increased. Over time, panel codes were replaced with higher order panel methods and subsequently CFD (Computational Fluid Dynamics). However, panel codes are still used for preliminary aerodynamic analysis as the time required for an analysis run is significantly less due to a decreased number of elements.

These are the various assumptions that go into developing potential flow panel methods:

• Inviscid
• Incompressible ${\displaystyle \nabla \cdot V=0}$
• Irrotational ${\displaystyle \nabla \times V=0}$
• Steady ${\displaystyle {\frac {\partial }{\partial t}}=0}$

However, the incompressible flow assumption may be removed from the potential flow derivation leaving:

• Potential flow (inviscid, irrotational, steady) ${\displaystyle \nabla ^{2}\phi =0}$

• From Small Disturbances

${\displaystyle (1-M_{\infty }^{2})\phi _{xx}+\phi _{yy}+\phi _{zz}=0}$ (subsonic)

• From Divergence Theorem

${\displaystyle \iiint \limits _{V}\left(\nabla \cdot \mathbf {F} \right)dV=\iint \limits _{S}\mathbf {F} \cdot \mathbf {n} \,dS}$

• Let Velocity U be a twice continuously differentiable function in a region of volume V in space. This function is the stream function ${\displaystyle \phi }$.
• Let P be a point in the volume V
• Let S be the surface boundary of the volume V.
• Let Q be a point on the surface S, and ${\displaystyle R=|P-Q|}$.

As Q goes from inside V to the surface of V,

• Therefore:

${\displaystyle U_{p}=-{\frac {1}{4\pi }}\iiint \limits _{V}\left({\frac {\nabla ^{2}\cdot \mathbf {U} }{R}}\right)dV_{Q}}$

${\displaystyle -{\frac {1}{4\pi }}\iint \limits _{S}\left({\frac {\mathbf {n} \cdot \nabla \mathbf {U} }{R}}\right)dS_{Q}}$

${\displaystyle +{\frac {1}{4\pi }}\iint \limits _{S}\left(\mathbf {U} \mathbf {n} \cdot \nabla {\frac {1}{R}}\right)dS_{Q}}$

For :${\displaystyle \nabla ^{2}\phi =0}$, where the surface normal points inwards.

${\displaystyle \phi _{p}=-{\frac {1}{4\pi }}\iint \limits _{S}\left(\mathbf {n} {\frac {\nabla \phi _{U}-\nabla \phi _{L}}{R}}-\mathbf {n} \left(\phi _{U}-\phi _{L}\right)\nabla {\frac {1}{R}}\right)dS_{Q}}$

This equation can be broken down into both a source term and a doublet term.

The Source Strength at an arbitrary point Q is:

${\displaystyle \sigma =\nabla \mathbf {n} (\nabla \phi _{U}-\nabla \phi _{L})}$

The Doublet Strength at an arbitrary point Q is:

${\displaystyle \mu =\phi _{U}-\phi _{L}}$

The simplified potential flow equation is:

${\displaystyle \phi _{p}=-{\frac {1}{4\pi }}\iint \limits _{S}\left({\frac {\sigma }{R}}-\mu \cdot \mathbf {n} \cdot \nabla {\frac {1}{R}}\right)dS}$

With this equation, along with applicable boundary conditions, the potential flow problem may be solved.

The velocity potential on the internal surface and all points inside V (or on the lower surface S) is 0.

${\displaystyle \phi _{L}=0}$

The Doublet Strength is:

${\displaystyle \mu =\phi _{U}-\phi _{L}}$

${\displaystyle \mu =\phi _{U}}$

The velocity potential on the outer surface is normal to the surface and is equal to the freestream velocity.

${\displaystyle \phi _{U}=-V_{\infty }\cdot \mathbf {n} }$

These basic equations are satisfied when the geometry is a 'watertight' geometry. If it is watertight, it is a well-posed problem. If it is not, it is an ill-posed problem.

The potential flow equation with well-posed boundary conditions applied is:

${\displaystyle \mu _{P}={\frac {1}{4\pi }}\iint \limits _{S}\left({\frac {V_{\infty }\cdot \mathbf {n} }{R}}\right)dS_{U}+{\frac {1}{4\pi }}\iint \limits _{S}\left(\mu \cdot \mathbf {n} \cdot \nabla {\frac {1}{R}}\right)dS}$

• Note that the ${\displaystyle dS_{U}}$ integration term is evaluated only on the upper surface, while th ${\displaystyle dS}$ integral term is evaluated on the upper and lower surfaces.

The continuous surface S may now be discretized into discrete panels. These panels will approximate the shape of the actual surface. This value of the various source and doublet terms may be evaluated at a convenient point (such as the centroid of the panel). Some assumed distribution of the source and doublet strengths (typically constant or linear) are used at points other than the centroid. A single source term s of unknown strength ${\displaystyle \lambda }$ and a single doublet term m of unknown strength ${\displaystyle \lambda }$ are defined at a given point.

${\displaystyle \sigma _{Q}=\sum _{i=1}^{n}\lambda _{i}s_{i}(Q)=0}$

${\displaystyle \mu _{Q}=\sum _{i=1}^{n}\lambda _{i}m_{i}(Q)}$

where:

${\displaystyle s_{i}=ln(r)}$

${\displaystyle m_{i}=}$

These terms can be used to create a system of linear equations which can be solved for all the unknown values of ${\displaystyle \lambda }$.

• constant strength - simple, large number of panels required
• linear varying strength - reasonable answer, little difficulty in creating well-posed problems
• quadratic varying strength - accurate, more difficult to create a well-posed problem

Some techniques are commonly used to model surfaces.[1]

• Body Thickness by line sources
• Body Lift by line doublets
• Wing Thickness by constant source panels
• Wing Lift by constant pressure panels
• Wing-Body Interface by constant pressure panels

Once the Velocity at every point is determined, the pressure can be determined by using one of the following formulas. All various Pressure coefficient methods produce results that are similar and are commonly used to identify regions where the results are invalid.

Pressure Coefficient is defined as:

${\displaystyle C_{p}={\frac {p-p_{\infty }}{q_{\infty }}}={\frac {p-p_{\infty }}{{\frac {1}{2}}\rho _{\infty }V_{\infty }^{2}}}={\frac {p-p_{\infty }}{{\frac {\gamma }{2}}p_{\infty }M_{\infty }^{2}}}}$

The Isentropic Pressure Coefficient is:

${\displaystyle C_{p}={\frac {2}{\gamma M_{\infty }^{2}}}\left(\left(1+{\frac {\gamma -1}{2}}M_{\infty }^{2}\left[{\frac {1-|{\vec {V}}|^{2}}{|{\vec {V_{\infty }}}|^{2}}}\right]\right)^{\frac {\gamma }{\gamma -1}}-1\right)}$

The Incompressible Pressure Coefficient is:

${\displaystyle C_{p}=1-{\frac {|{\vec {V}}|^{2}}{|{\vec {V_{\infty }}}|^{2}}}}$

The Second Order Pressure Coefficient is:

${\displaystyle C_{p}=1-|{\vec {V}}|^{2}+M_{\infty }^{2}u^{2}}$

The Slender Body Theory Pressure Coefficient is:

${\displaystyle C_{p}=-(2u+v^{2}+w^{2})}$

The Linear Theory Pressure Coefficient is:

${\displaystyle C_{p}=-2u}$

The Reduced Second Order Pressure Coefficient is:

${\displaystyle C_{p}=1-|{\vec {V}}|^{2}}$

• Panel methods are inviscid solutions. You will not capture viscous effects except via user "modeling" by changing the geometry.
• Solutions are invalid as soon as the flow changes locally from subsonic to supersonic (i.e. the critical Mach number has been exceeded) or vice versa.

• Stream function
• Conformal mapping
• Velocity potential
• Divergence theorem
• Joukowsky transform
• Potential flow
• Circulation
• Biot–Savart law

• Public Domain Aerodynamic Software, A Panair Distribution Source, Ralph Carmichael
• Panair Volume I, Theory Manual, Version 3.0, Michael Epton, Alfred Magnus, 1990 Boeing
• Panair Volume II, Theory Manual, Version 3.0, Michael Epton, Alfred Magnus, 1990 Boeing
• Panair Volume III, Case Manual, Version 1.0, Michael Epton, Kenneth Sidewell, Alfred Magnus, 1981 Boeing
• Panair Volume IV, Maintenance Document, Version 3.0, Michael Epton, Kenneth Sidewell, Alfred Magnus, 1991 Boeing
• Recent Experience in Using Finite Element Methods For The Solution Of Problems In Aerodynamic Interference, Ralph Carmichael, 1971 NASA Ames Research Center